An asymptote is when the value of the graph is undefined, we know that something is undefined when it is divided by zero. Now let's take a look at our ratio identities tan is equal to y/x, cot equal to x/y, secant is r/x and cosecant is r/y, if we look at al of these we notice that these mean the numerator are divided by a number ;x, y these could be any number including 0. Meaning that any one of these could have an asymptote. If we look at cos and sin their ratio identities are y/r and x/r r will always equal one and therefore these will never be divided by 0, they won't have an asymptote.
Tan= y/x Sin= y/r
Cot= x/y Cos= x/r
Sec= r/x
Csc= r/y
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